Lemma 7 in [1]} shows that the unique minimizer of
\(h(y) = c(x,y)-\hat{\phi }^c(y)\)  is achieved at \(y_*=\exp _x[-\nabla _x\hat{\phi }(x)]\) . In particular, \(\nabla _x\hat{\phi }(x) = -\log _x(y_\star )\) . As explained above, \(y_*\)  is not on the cut locus of \(x\) .
